As determining the presence of an object is a quantum state discrimination problem it is worthwhile to mention that an optimal measurement exists. While the exact optimal measurement is unknown because its POVM is, in general, difficult to calculate and realise, calculable bounds exist[1]}. The bounds demonstrate that the quantum illumination states are more distinguishable than the corresponding states from a classical illumination-based LIDAR system[2]}. Following on from the discussion on POVMs in Section , we extend this to include other aspects of our model. As our target object and our inefficient detector models each include an attenuation parameter (reflectivity or quantum efficiency) and a thermal background (the target environment and the detector dark noise) we use a combined POVM for the reflection and detection that accounts for these aspects of our LIDAR model: sub-optimal object reflectivity, environmental background noise, and detector inefficiencies. Then probabilities of click and no-click events on the detectors for both classical and quantum illumination is all of the information we can calculate in this simple detection LIDAR scheme.
